Chapter 21, Problem 7PS

Summary Introduction

Call option:

A call option is an agreement where the buyer is entitled to a right to buy a stock at a pre-specified price (known as exercise price or strike price) within a pre-specified period. The stock on which the call option is provided is called the underlying asset.

Put option:

It is an agreement where the buyer is entitled to a right to sell a stock at a pre-specified price (known as exercise price or strike price) within a pre-specified period. The stock on which the put option is provided is called the underlying asset.

Hedge ratio:

A hedge ratio indicates the level of exposure of an investment to risk. For example, if a hedge ratio for an investment is 0.60 or 60%, it means 60% of that investment is protected from risk and remaining 40% of the investment is exposed to risk. A hedge ratio is calculated by dividing the option price range by the stock price range.

To compute:

The hedge ratio when the exercise price is

1. $120

Determine the impact of more in the money on hedge ratio.

Answer to Problem 7PS

The hedge ratio is

0

When the option becomes progressively more in the money, the value of the hedge ratio increases. It indicates that it is possible to protect the investment more, when the option becomes progressively more in the money.

Explanation of Solution

(a)

In the given case, stock price is $100. Up parameter (u) is 1.2 and down (d) is 0.9. So, after one year stock price may be either $\$120\left(=uS=\$100\times 1.2\right)$ or $\$90\left(=dS=\$100\times 0.90\right)$ . After one year, the up value of call will be $\$0\left(=C_u=\$120-\$120\right)$ and the down value of call will be $\$0\left(=C_d=\$90-\$120\right)$ , considering the exercise price of $120.

Given:

  $\begin{array}{l}{C}_u=0\\ C_d=0\\ uS=\$120\\ dS=\$90\end{array}$

Calculation:

$\begin{array}{c}Hedgeratio=\frac{C_u-C_d}{uS-dS}\\ =\frac{0}{\$120-\$90}\\ =0\end{array}$

Summary Introduction

(b)

Call option:

A call option is an agreement that gives the buyer the right to buy a stock at a pre-specified price within a pre-specified period. The stock on which the call option is provided is called the underlying asset.

Put option:

A put option is an agreement that gives the buyer the right to sell a stock at a pre-specified price within a pre-specified period. The stock on which the put option is provided is called the underlying asset.

Hedge ratio:

A hedge ratio indicates the level of exposure of an investment to risk. For example, if a hedge ratio for an investment is 0.60 or 60%, it means 60% of that investment is protected from risk and remaining 40% of the investment is exposed to risk. A hedge ratio is calculated by dividing the option price range by the stock price range.

To compute:

The hedge ratio when the exercise price is

$110

Determine the impact of more in the money on hedge ratio.

Answer to Problem 7PS

The hedge ratio is

0.33

Explanation of Solution

In the given case, stock price is $100. Up parameter (u) is 1.2 and down (d) is 0.9. So, after one year stock price may be either $\$120\left(=uS=\$100\times 1.2\right)$ or $\$90\left(=dS=\$100\times 0.90\right)$ . After one year, the up value of call will be $\$10\left(=C_u=\$120-\$110\right)$ and the down value of call will be $\$0\left(=C_d=\$90-\$110\right)$ , considering the exercise price of $110.

Given:

  $\begin{array}{l}{C}_u=\$10\\ C_d=0\\ uS=\$120\\ dS=\$90\end{array}$

Calculation:

$\begin{array}{c}Hedgeratio=\frac{C_u-C_d}{uS-dS}\\ =\frac{\$10-0}{\$120-\$90}\\ =0.333\end{array}$

Summary Introduction

(C)

Call option:

A call option is an agreement that gives the buyer the right to buy a stock at a pre-specified price within a pre-specified period. The stock on which the call option is provided is called the underlying asset.

Put option:

A put option is an agreement that gives the buyer the right to sell a stock at a pre-specified price within a pre-specified period. The stock on which the put option is provided is called the underlying asset.

Hedge ratio:

A hedge ratio indicates the level of exposure of an investment to risk. For example, if a hedge ratio for an investment is 0.60 or 60%, it means 60% of that investment is protected from risk and remaining 40% of the investment is exposed to risk. A hedge ratio is calculated by dividing the option price range by the stock price range.

To compute:

The hedge ratio when the exercise price is

$100

Determine the impact of more in the money on hedge ratio.

Answer to Problem 7PS

The hedge ratio is

0.67

When the option becomes progressively more in the money, the value of the hedge ratio increases. It indicates that it is possible to protect the investment more, when the option becomes progressively more in the money.

Explanation of Solution

In the given case, stock price is $100. Up parameter (u) is 1.2 and down (d) is 0.9. So, after one year stock price may be either $\$120\left(=uS=\$100\times 1.2\right)$ or $\$90\left(=dS=\$100\times 0.90\right)$ . After one year, the up value of call will be $\$20\left(=C_u=\$120-\$100\right)$ and the down value of call will be $\$0\left(=C_d=\$90-\$100\right)$ , considering the exercise price of $100.

Given:

  $\begin{array}{l}{C}_u=\$20\\ C_d=0\\ uS=\$120\\ dS=\$90\end{array}$

Calculation:

$\begin{array}{c}Hedgeratio=\frac{C_u-C_d}{uS-dS}\\ =\frac{\$20-0}{\$120-\$90}\\ =0.67\end{array}$

Summary Introduction

(D)

Call option:

A call option is an agreement that gives the buyer the right to buy a stock at a pre-specified price within a pre-specified period. The stock on which the call option is provided is called the underlying asset.

Put option:

A put option is an agreement that gives the buyer the right to sell a stock at a pre-specified price within a pre-specified period. The stock on which the put option is provided is called the underlying asset.

Hedge ratio:

A hedge ratio indicates the level of exposure of an investment to risk. For example, if a hedge ratio for an investment is 0.60 or 60%, it means 60% of that investment is protected from risk and remaining 40% of the investment is exposed to risk. A hedge ratio is calculated by dividing the option price range by the stock price range.

To compute:

The hedge ratio when the exercise price is

1. $120
2. $110
3. $100
4. $90 and

Determine the impact of more in the money on hedge ratio.

Answer to Problem 7PS

1

When the option becomes progressively more in the money, the value of the hedge ratio increases. It indicates that it is possible to protect the investment more, when the option becomes progressively more in the money.

Explanation of Solution

In the given case, stock price is $100. Up parameter (u) is 1.2 and down (d) is 0.9. So, after one year stock price may be either $\$120\left(=uS=\$100\times 1.2\right)$ or $\$90\left(=dS=\$100\times 0.90\right)$ . After one year, the up value of call will be $\$30\left(=C_u=\$120-\$90\right)$ and the down value of call will be $\$0\left(=C_d=\$90-\$90\right)$ , considering the exercise price of $90.

Given:

  $\begin{array}{l}{C}_u=\$30\\ C_d=0\\ uS=\$120\\ dS=\$90\end{array}$

Calculation:

$\begin{array}{c}Hedgeratio=\frac{C_u-C_d}{uS-dS}\\ =\frac{\$30-0}{\$120-\$90}\\ =1\end{array}$